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What is the value of F(-1)-F(1)?

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What is the value of F(-1)-F(1)?  [#permalink]

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New post 10 Feb 2019, 19:13
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GMATH practice exercise (Quant Class 14)

What is the value of F(-1)-F(1)?

(1) F(x) = x^2 , for all x
(2) F(x+1) = F(x) + 2x + 1, for all x

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Re: What is the value of F(-1)-F(1)?  [#permalink]

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New post 10 Feb 2019, 19:40
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

What is the value of F(-1)-F(1)?

(1) F(x) = x^2 , for all x
(2) F(x+1) = F(x) + 2x + 1, for all x



So we are looking for \(F(-1)-F(1)\).

(1) \(F(x) = x^2\) , for all x
We can get the value of F(-1) and F(1), so sufficient
\(F(-1) = (-1)^2=1\)
\(F(1) = (1)^2=1\)
so \(F(-1)-F(1)=1-1=0\)
Sufficient

(2) \(F(x+1) = F(x) + 2x + 1\), for all x
The difference of 1 in x and x+1 should tempt you to rearrange and work on the variables accordingly..
Now F(-1) and F(1) have difference of 1-(-1) or 2, so let us work in similar fashion..
\(F(x+1) = F(x) + 2x + 1......... F(x)-F(x+1) = -( 2x + 1)\)
a) let x = -1, so \(F(x)-F(x+1) = -( 2x + 1)......F(-1)-F(-1+1) = -( 2(-1) + 1)=-(-1)=1\), that is F(-1)-F(0) = 1
b) let x = 0, so \(F(x)-F(x+1) = -( 2x + 1)......F(0)-F(0+1) = -( 2(0) + 1)=-(1)=-1\), that is F(0)-F(1) = -1
Add the two colored equation F(-1)-F(0)+ F(0)-F(1) = 1+ (-1)..........
so \(F(-1)-F(1)=1-1=0\)
Sufficient

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Re: What is the value of F(-1)-F(1)?  [#permalink]

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New post 10 Feb 2019, 22:58
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

What is the value of F(-1)-F(1)?

(1) F(x) = x^2 , for all x
(2) F(x+1) = F(x) + 2x + 1, for all x



This approach is really good and I got struck at the highlighted point.

Here is my approach.The main concern is statement 2

1.F(x) = x^2 , for all x

F(-1) = x^2 = 1
F(1)= x^2 = 1

So we have F(-1) - F(1)= 0


2.
F(x+1) = F(x) + 2x + 1, for all x ---------------------- Main equation

Now my line of thinking was:Lets get F(-1) so I did x+1=-1 which gave x=-2.
Now substituting in main equation 2

F(-2+1)= F(-2) + 2*(-2) +1

=> F(-1) = F(-2) - 4 + 1
=F(-2) -3

I think this is wrong step,I identified a little later.


We should rather take X=-1 directlly ,this will save us from getting F(-2) and have equation in F(0) form thereby saving us from other variable.

so substituing X=-1 in main equation

F(0) = F(-1) + 2*-1 +1
= F(-1) - 1 ---------------------------------- 1

Again to get F(1) Lets make x+1= 1 ,X=0
Now again substituting in main equation 2

F(1) = F(0) + 2*0 + 1
=F(0) + 1 --------------------2
implies

F(0) = F(1)-1

Substituting F(0) in equation 1

F(1)-1 = F(-1) - 1

Rearranging

F(-1) - F(1) = 0

Hence sufficient.



Hope it helps!!
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Re: What is the value of F(-1)-F(1)?  [#permalink]

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New post 11 Feb 2019, 10:28
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

What is the value of F(-1)-F(1)?

(1) F(x) = x^2 , for all x
(2) F(x+1) = F(x) + 2x + 1, for all x

\(? = F\left( { - 1} \right) - F\left( 1 \right)\)

\(\left( 1 \right)\,\,F\left( x \right) = {x^2}\,,\,\,{\rm{for}}\,\,{\rm{all}}\,\,x\,\,\,\, \Rightarrow \,\,\,\,F\left( x \right) = F\left( { - x} \right)\,,\,\,{\rm{for}}\,\,{\rm{all}}\,\,x\,\,\,\,\mathop \Rightarrow \limits^{x\, = \,1} \,\,\,\,\,F\left( 1 \right) = F\left( { - 1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 0\)


\(\left( 2 \right)\,\,F\left( {x + 1} \right) - F\left( x \right) = 2x + 1\,,\,{\rm{for}}\,\,{\rm{all}}\,\,x\,\,\,\left( * \right)\)

\(\left. \matrix{
{\rm{Take}}\,\,x = - 1\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,F\left( 0 \right) - F\left( { - 1} \right) = - 2 + 1\,\,\, \hfill \cr
{\rm{Take}}\,\,x = 0\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,F\left( 1 \right) - F\left( 0 \right) = 1 \hfill \cr} \right\}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,F\left( 1 \right) - F\left( { - 1} \right) = 1 + \left( { - 2 + 1} \right) = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,? = - 0 = 0\)


The correct answer is therefore (D).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: What is the value of F(-1)-F(1)?   [#permalink] 11 Feb 2019, 10:28
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